Mooney-Rivlin constants

In order to check this prediction, stress-strain measurements were made up to moderate strains at room temperature. The obtained data are plotted in the usual manner as a versus 1/X in Figure 8. Table V gives the Mooney-Rivlin constants 2C and 2C calculated from these plots and also the ratio C./Cj. 

Mooney-Rivlin constants obtained from strain dependent measurements at 298 K... 

Figure 9. Dependence of the Mooney-Rivlin constant on extent of cross-linking. Mn is 21,600 PDMS. Key , 2C, X, 2CS (3).

In this contribution, we report equilibrium modulus and sol fraction measurements on diepoxidet-monoepoxide-diamine networks and polyoxypropylene triol-diisocyanate networks and a comparison with calculated values. A practically zero (epoxides) or low (polyurethanes) Mooney-Rivlin constant C and a low and accounted for wastage of bonds in elastically inactive cycles are the advantages of the systems. Plots of reduced modulus against the gel fraction have been used, because they have been found to minimize the effect of EIC, incompleteness of the reaction, or possible errors in analytical characteristics (16-20). A full account of the work on epoxy and polyurethane networks including the statistical derivation of various structural parameters will be published separately elsewhere. 

The phantom network behaviour corresponding to volumeless chains which can freely interpenetrate one through the other and thus to unrestricted fluctuations of crosslinks should be approached in swollen systems or at high strains (proportionality to the Mooney-Rivlin constant C-j). For suppressed fluctuations of crosslinks, which then are displaced affinely with the strain, A for the small-strain modulus (equal to C1+C2) approaches unity. This situation should be characteristic of bulk systems. The constraints arising from interchain interactions important at low strains should be reflected in an increase of Aabove the phantom value and no extra Gee contribution to the modulus is necessary. The upper limit of the predicted equilibrium modulus corresponds therefore, A = 1. 

The results of stress-strain measurements can be summarized as follows (1) the reduced stress S (A- A ) (Ais the extension ratio) is practically independent of strain so that the Mooney-Rivlin constant C2 is practically zero for dry as well as swollen samples (C2/C1=0 0.05) (2) the values of G are practically the same whether obtained on dry or swollen samples (3) assuming that Gee=0, the data are compatible with the chemical contribution and A 1 (4) the difference between the phantom network dependence with the value of A given by Eq.(4) and the experimental moduli fits well the theoretical dependence of G e in Eq.(2) or (3). The proportionality constant in G for series of networks with s equal to 0, 0.2, 0.33, and 0. Ewas practically the same -(8.2, 6.3, 8.8, and 8.5)x10-4 mol/cm with the average value 7.95x10 mol/cm. Results (1) and (2) suggest that phantom network behavior has been reached, but the result(3) is contrary to that. Either the constraints do survive also in the swollen and stressed states, or we have to consider an extra contribution due to the incrossability of "phantom" chains. The latter explanation is somewhat supported by the constancy of in Eq.(2) for a series of samples of different composition. 

Again it has been found experimentally that (Cx + C2) does not correspond with the Mooney-Rivlin constants in extension and compression. One is thus faced with the problem of deciding which Cx to identify with the Gaussian constant. Several authors consider Ct in extension to be the Gaussian constant, but in view of the above this position cannot be maintained. One could instead identify the compression modulus with the Gaussian constant 3(AvkT/Lt) Kr2)4/(r2 0) because in compression Cg is always found to be very small or zero. There is, however, no theoretical justification for this procedure either. 

Figure 11-7. Ratio Cil C of the Mooney-Rivlin constants of different elastomers as a function of the chain cross section for 2Ci = 0.2 MPa. (After R. F. Boyer and R. L. Miller.)...

The phenomenological Mooney-Rivlin constant Cj is essentially independent of... 

In many cases it is reasonable to take the initial value of [f ]ph = 2Ci, where 2Ci is the first Mooney-Rivlin constant. An alternative possibility is to estimate [f ]ph from the stoichiometry of the chemical reaction using Eqs. (29.12)-(29.14) and (29.41). 

Specimen MJIO (glmol) (LALLS) (glmol) (GPC) V from Mooney-Rivlin constant (10 moljg) V from equilibrium swelling (10 moljg)... 

Figure 6. A linear fit showing a correlation of stiffness (represented by the Mooney-Rivlin constant Cl) and d-spacing.

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